Thus we see that \(\Delta f(x) = \operatorname{tr}(\mathrm {Hess}_f)(x)\)  for \(\mathcal {H}^5\) -a.e. \(x \in X_{\epsilon }\) . Then it follows from the Bochner inequality involving the Hessian term proved in [1]} that in the weak sense:

12|f|2 |Hessf|2 +f, f+K1|f|2
